A block of mass \(m\) is placed on a surface with a vertical cross section given by \(y=\dfrac{x^3}{6}.\) If the coefficient of friction is \(0.5,\) then the maximum height above the ground at which the block can be placed without slipping is:
1. \( \dfrac{2}{3}~\text{m} \)
2. \( \dfrac{1}{3} ~\text{m} \)
3. \( \dfrac{1}{2} ~\text{m} \)
4. \( \dfrac{1}{6} ~\text{m}\)

Given in the figure are two blocks \(A\) and \(B\) of weight \(20~\text{N}\) and \(100~\text{N}\), respectively. These are being pressed against a wall by a force \(F\) as shown. If the coefficient of friction between the blocks is \(0.1\) and between block \(B\) and the wall is \(0.15\), the frictional force applied by the wall on block \(B\) is:
       
1. \(100~\text{N}\)
2. \(80~\text{N}\)
3. \(120~\text{N}\)
4. \(150~\text{N}\)

Two masses \(m_1=5~\text{kg}\) and \(m_2=10~\text{kg}\), connected by an inextensible string over a frictionless pulley, are moving as shown in the figure. The coefficient of friction of the horizontal surface is \(0.15\). The minimum weight \(m\) that should be put on top of \(m_2\) to stop the motion is:

1. \(18.3~\text{kg}\)
2. \(23.3~\text{kg}\)
3. \(43.3~\text{kg}\)
4. \(10.3~\text{kg}\)

Two blocks \(A\) and \(B\) of masses \(m_A=1~\text{kg}\) and \(m_B=3~\text{kg}\) are kept on the table as shown in figure. the coefficient of friction between \(A\) and \(B\) is \(0.2\) and between \(B\) and the surface of the table is also \(0.2\). The maximum force \(F\) that can be applied on \(B\) horizontally, so that the block \(A\) does not slide over the block \(B\) is: [Take \(g= 10\) m/s²]

                     
1. \(8~\text{N}\)
2. \(40~\text{N}\)
3. \(16~\text{N}\)
4. \(12~\text{N}\)

A block of mass \(5\) kg is (i) pushed in case (\(A\)) and (ii) pulled in case (\(B\)), by a force \(F = 20~\text{N}\), making an angle of \(30^\circ\) with the horizontal, as shown in the figures. The coefficient of friction between the block and floor is \(\mu=0.2\). The difference between the accelerations of the blocks, in case (\(B\)) and case (\(A\)) will be: (\(g=10\) ms^–2)

1. \(3.2~\text{ms}^{-2}\)
2. \(0.8~\text{ms}^{-2}\)
3. \(0~\text{ms}^{-1}\)
4. \(0.4~\text{ms}^{-1}\)

An insect is at the bottom of a hemispherical ditch of radius \(1~\text{m}\). It crawls up the ditch but starts slipping after it is at height \(h\) from the bottom. If the coefficient of friction between the ground and the insect is \(0.75\), then \(h\) is: \(\left(g = 10~\text{ms}^{-2}\right)\)
1. \(0.20~\text{m}\)
2. \(0.60~\text{m}\)
3. \(0.45~\text{m}\)
4. \(0.80~\text{m}\)

The coefficient of static friction between a wooden block of mass \(0.5\) kg and a vertical rough wall is \(0.2\). The magnitude of horizontal force that should be applied on the block to keep it adhere to the wall will be:
($\mathrm{}$Take \(g=10~\mathrm{ms^{-2}}\) )
1. \(10~\text{N}\)
2. \(25~\text{N}\)
3. \(5~\text{N}\)
4. \(30~\text{N}\)

An inclined plane is bent in such a way that the vertical cross-section is given by, \(y=\frac{x^2}{4}\) where \(y\) is in vertical and \(x\) in the horizontal direction. If the upper surface of this curved plane is rough with a coefficient of friction \(\mu=0.5\), the maximum height in(cm) at which a stationary block will not slip downward is:
1. \(25\)
2. \(50\)
3. \(75\)
4. \(100\)